1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| A.Asperti, C.Sacerdoti Coen, *)
8 (* ||A|| E.Tassi, S.Zacchiroli *)
10 (* \ / Matita is distributed under the terms of the *)
11 (* v GNU Lesser General Public License Version 2.1 *)
13 (**************************************************************************)
15 set "baseuri" "cic:/matita/nat/primes".
17 include "nat/div_and_mod.ma".
18 include "nat/minimization.ma".
19 include "nat/sigma_and_pi.ma".
20 include "nat/factorial.ma".
22 inductive divides (n,m:nat) : Prop \def
23 witness : \forall p:nat.m = times n p \to divides n m.
25 theorem reflexive_divides : reflexive nat divides.
28 exact witness x x (S O) (times_n_SO x).
31 theorem divides_to_div_mod_spec :
32 \forall n,m. O < n \to divides n m \to div_mod_spec m n (div m n) O.
33 intros.elim H1.rewrite > H2.
34 constructor 1.assumption.
35 apply lt_O_n_elim n H.intros.
37 rewrite > div_times.apply sym_times.
40 theorem div_mod_spec_to_div :
41 \forall n,m,p. div_mod_spec m n p O \to divides n m.
45 rewrite > plus_n_O (p*n).assumption.
48 theorem divides_to_mod_O:
49 \forall n,m. O < n \to divides n m \to (mod m n) = O.
50 intros.apply div_mod_spec_to_eq2 m n (div m n) (mod m n) (div m n) O.
51 apply div_mod_spec_div_mod.assumption.
52 apply divides_to_div_mod_spec.assumption.assumption.
55 theorem mod_O_to_divides:
56 \forall n,m. O< n \to (mod m n) = O \to divides n m.
58 apply witness n m (div m n).
59 rewrite > plus_n_O (n*div m n).
62 (* Andrea: perche' hint non lo trova ?*)
67 theorem divides_n_O: \forall n:nat. divides n O.
68 intro. apply witness n O O.apply times_n_O.
71 theorem divides_SO_n: \forall n:nat. divides (S O) n.
72 intro. apply witness (S O) n n. simplify.apply plus_n_O.
75 theorem divides_plus: \forall n,p,q:nat.
76 divides n p \to divides n q \to divides n (p+q).
78 elim H.elim H1. apply witness n (p+q) (n2+n1).
79 rewrite > H2.rewrite > H3.apply sym_eq.apply distr_times_plus.
82 theorem divides_minus: \forall n,p,q:nat.
83 divides n p \to divides n q \to divides n (p-q).
85 elim H.elim H1. apply witness n (p-q) (n2-n1).
86 rewrite > H2.rewrite > H3.apply sym_eq.apply distr_times_minus.
89 theorem divides_times: \forall n,m,p,q:nat.
90 divides n p \to divides m q \to divides (n*m) (p*q).
92 elim H.elim H1. apply witness (n*m) (p*q) (n2*n1).
93 rewrite > H2.rewrite > H3.
94 apply trans_eq nat ? (n*(m*(n2*n1))).
95 apply trans_eq nat ? (n*(n2*(m*n1))).
98 apply trans_eq nat ? ((n2*m)*n1).
99 apply sym_eq. apply assoc_times.
100 rewrite > sym_times n2 m.apply assoc_times.
101 apply sym_eq. apply assoc_times.
104 theorem transitive_divides: \forall n,m,p.
105 divides n m \to divides m p \to divides n p.
107 elim H.elim H1. apply witness n p (n2*n1).
108 rewrite > H3.rewrite > H2.
113 theorem divides_to_le : \forall n,m. O < m \to divides n m \to n \le m.
114 intros. elim H1.rewrite > H2.cut O < n2.
115 apply lt_O_n_elim n2 Hcut.intro.rewrite < sym_times.
116 simplify.rewrite < sym_plus.
118 elim le_to_or_lt_eq O n2.
120 absurd O<m.assumption.
121 rewrite > H2.rewrite < H3.rewrite < times_n_O.
126 theorem divides_to_lt_O : \forall n,m. O < m \to divides n m \to O < n.
128 elim le_to_or_lt_eq O n (le_O_n n).
130 rewrite < H3.absurd O < m.assumption.
131 rewrite > H2.rewrite < H3.
132 simplify.exact not_le_Sn_n O.
135 (* boolean divides *)
136 definition divides_b : nat \to nat \to bool \def
137 \lambda n,m :nat. (eqb (mod m n) O).
139 theorem divides_b_to_Prop :
140 \forall n,m:nat. O < n \to
141 match divides_b n m with
142 [ true \Rightarrow divides n m
143 | false \Rightarrow \lnot (divides n m)].
146 match eqb (mod m n) O with
147 [ true \Rightarrow divides n m
148 | false \Rightarrow \lnot (divides n m)].
150 intro.simplify.apply mod_O_to_divides.assumption.assumption.
151 intro.simplify.intro.apply H1.apply divides_to_mod_O.assumption.assumption.
154 theorem divides_b_true_to_divides :
155 \forall n,m:nat. O < n \to
156 (divides_b n m = true ) \to divides n m.
160 [ true \Rightarrow divides n m
161 | false \Rightarrow \lnot (divides n m)].
162 rewrite < H1.apply divides_b_to_Prop.
166 theorem divides_b_false_to_not_divides :
167 \forall n,m:nat. O < n \to
168 (divides_b n m = false ) \to \lnot (divides n m).
172 [ true \Rightarrow divides n m
173 | false \Rightarrow \lnot (divides n m)].
174 rewrite < H1.apply divides_b_to_Prop.
178 theorem decidable_divides: \forall n,m:nat.O < n \to
179 decidable (divides n m).
180 intros.change with (divides n m) \lor \not (divides n m).
182 match divides_b n m with
183 [ true \Rightarrow divides n m
184 | false \Rightarrow \not (divides n m)] \to (divides n m) \lor \not (divides n m).
185 apply Hcut.apply divides_b_to_Prop.assumption.
186 elim (divides_b n m).left.apply H1.right.apply H1.
189 theorem divides_to_divides_b_true : \forall n,m:nat. O < n \to
190 divides n m \to divides_b n m = true.
192 cut match (divides_b n m) with
193 [ true \Rightarrow (divides n m)
194 | false \Rightarrow \not (divides n m)] \to ((divides_b n m) = true).
195 apply Hcut.apply divides_b_to_Prop.assumption.
196 elim divides_b n m.reflexivity.
197 absurd (divides n m).assumption.assumption.
200 theorem not_divides_to_divides_b_false: \forall n,m:nat. O < n \to
201 \not(divides n m) \to (divides_b n m) = false.
203 cut match (divides_b n m) with
204 [ true \Rightarrow (divides n m)
205 | false \Rightarrow \not (divides n m)] \to ((divides_b n m) = false).
206 apply Hcut.apply divides_b_to_Prop.assumption.
208 absurd (divides n m).assumption.assumption.
213 theorem divides_f_pi_f : \forall f:nat \to nat.\forall n,i:nat.
214 i < n \to divides (f i) (pi n f).
215 intros 3.elim n.apply False_ind.apply not_le_Sn_O i H.
217 apply le_n_Sm_elim (S i) n1 H1.
219 apply transitive_divides ? (pi n1 f).
220 apply H.simplify.apply le_S_S_to_le. assumption.
221 apply witness ? ? (f n1).apply sym_times.
224 apply witness ? ? (pi n1 f).reflexivity.
225 apply inj_S.assumption.
228 theorem mod_S_pi: \forall f:nat \to nat.\forall n,i:nat.
229 i < n \to (S O) < (f i) \to mod (S (pi n f)) (f i) = (S O).
230 intros.cut mod (pi n f) (f i) = O.
232 apply mod_S.apply trans_lt O (S O).apply le_n (S O).assumption.
233 rewrite > Hcut.assumption.
234 apply divides_to_mod_O.apply trans_lt O (S O).apply le_n (S O).assumption.
235 apply divides_f_pi_f.assumption.
238 (* divides and fact *)
239 theorem divides_fact : \forall n,i:nat.
240 O < i \to i \le n \to divides i (fact n).
241 intros 3.elim n.absurd O<i.assumption.apply le_n_O_elim i H1.
243 change with divides i ((S n1)*(fact n1)).
244 apply le_n_Sm_elim i n1 H2.
246 apply transitive_divides ? (fact n1).
247 apply H1.apply le_S_S_to_le. assumption.
248 apply witness ? ? (S n1).apply sym_times.
251 apply witness ? ? (fact n1).reflexivity.
254 theorem mod_S_fact: \forall n,i:nat.
255 (S O) < i \to i \le n \to mod (S (fact n)) i = (S O).
256 intros.cut mod (fact n) i = O.
258 apply mod_S.apply trans_lt O (S O).apply le_n (S O).assumption.
259 rewrite > Hcut.assumption.
260 apply divides_to_mod_O.apply trans_lt O (S O).apply le_n (S O).assumption.
261 apply divides_fact.apply trans_lt O (S O).apply le_n (S O).assumption.
265 theorem not_divides_S_fact: \forall n,i:nat.
266 (S O) < i \to i \le n \to \not (divides i (S (fact n))).
268 apply divides_b_false_to_not_divides.
269 apply trans_lt O (S O).apply le_n (S O).assumption.
270 change with (eqb (mod (S (fact n)) i) O) = false.
271 rewrite > mod_S_fact.simplify.reflexivity.
272 assumption.assumption.
276 definition prime : nat \to Prop \def
277 \lambda n:nat. (S O) < n \land
278 (\forall m:nat. divides m n \to (S O) < m \to m = n).
280 theorem not_prime_O: \lnot (prime O).
281 simplify.intro.elim H.apply not_le_Sn_O (S O) H1.
284 theorem not_prime_SO: \lnot (prime (S O)).
285 simplify.intro.elim H.apply not_le_Sn_n (S O) H1.
288 (* smallest factor *)
289 definition smallest_factor : nat \to nat \def
295 [ O \Rightarrow (S O)
296 | (S q) \Rightarrow min_aux q (S(S q)) (\lambda m.(eqb (mod (S(S q)) m) O))]].
299 theorem example1 : smallest_prime_factor (S(S(S O))) = (S(S(S O))).
300 normalize.reflexivity.
303 theorem example2: smallest_prime_factor (S(S(S(S O)))) = (S(S O)).
304 normalize.reflexivity.
307 theorem example3 : smallest_prime_factor (S(S(S(S(S(S(S O))))))) = (S(S(S(S(S(S(S O))))))).
308 simplify.reflexivity.
311 theorem lt_SO_smallest_factor:
312 \forall n:nat. (S O) < n \to (S O) < (smallest_factor n).
314 apply nat_case n.intro.apply False_ind.apply not_le_Sn_O (S O) H.
315 intro.apply nat_case m.intro. apply False_ind.apply not_le_Sn_n (S O) H.
318 S O < min_aux m1 (S(S m1)) (\lambda m.(eqb (mod (S(S m1)) m) O)).
319 apply lt_to_le_to_lt ? (S (S O)).
321 cut (S(S O)) = (S(S m1)) - m1.
324 apply sym_eq.apply plus_to_minus.apply le_S.apply le_n_Sn.
325 rewrite < sym_plus.simplify.reflexivity.
328 theorem lt_O_smallest_factor: \forall n:nat. O < n \to O < (smallest_factor n).
330 apply nat_case n.intro.apply False_ind.apply not_le_Sn_n O H.
331 intro.apply nat_case m.intro.
333 intros.apply trans_lt ? (S O).
334 simplify. apply le_n.
335 apply lt_SO_smallest_factor.simplify. apply le_S_S.
336 apply le_S_S.apply le_O_n.
339 theorem divides_smallest_factor_n :
340 \forall n:nat. O < n \to divides (smallest_factor n) n.
342 apply nat_case n.intro.apply False_ind.apply not_le_Sn_O O H.
343 intro.apply nat_case m.intro. simplify.
344 apply witness ? ? (S O). simplify.reflexivity.
346 apply divides_b_true_to_divides.
347 apply lt_O_smallest_factor ? H.
349 eqb (mod (S(S m1)) (min_aux m1 (S(S m1))
350 (\lambda m.(eqb (mod (S(S m1)) m) O)))) O = true.
351 apply f_min_aux_true.
352 apply ex_intro nat ? (S(S m1)).
354 apply le_minus_m.apply le_n.
355 rewrite > mod_n_n.reflexivity.
356 apply trans_lt ? (S O).apply le_n (S O).simplify.
357 apply le_S_S.apply le_S_S.apply le_O_n.
360 theorem le_smallest_factor_n :
361 \forall n:nat. smallest_factor n \le n.
362 intro.apply nat_case n.simplify.reflexivity.
363 intro.apply nat_case m.simplify.reflexivity.
364 intro.apply divides_to_le.
365 simplify.apply le_S_S.apply le_O_n.
366 apply divides_smallest_factor_n.
367 simplify.apply le_S_S.apply le_O_n.
370 theorem lt_smallest_factor_to_not_divides: \forall n,i:nat.
371 (S O) < n \to (S O) < i \to i < (smallest_factor n) \to \lnot (divides i n).
373 apply nat_case n.intro.apply False_ind.apply not_le_Sn_O (S O) H.
374 intro.apply nat_case m.intro. apply False_ind.apply not_le_Sn_n (S O) H.
376 apply divides_b_false_to_not_divides.
377 apply trans_lt O (S O).apply le_n (S O).assumption.
378 change with (eqb (mod (S(S m1)) i) O) = false.
379 apply lt_min_aux_to_false
380 (\lambda i:nat.eqb (mod (S(S m1)) i) O) (S(S m1)) m1 i.
381 cut (S(S O)) = (S(S m1)-m1).
382 rewrite < Hcut.exact H1.
383 apply sym_eq. apply plus_to_minus.
384 apply le_S.apply le_n_Sn.
385 rewrite < sym_plus.simplify.reflexivity.
389 theorem prime_smallest_factor_n :
390 \forall n:nat. (S O) < n \to prime (smallest_factor n).
391 intro. change with (S(S O)) \le n \to (S O) < (smallest_factor n) \land
392 (\forall m:nat. divides m (smallest_factor n) \to (S O) < m \to m = (smallest_factor n)).
394 apply lt_SO_smallest_factor.assumption.
396 cut le m (smallest_factor n).
397 elim le_to_or_lt_eq m (smallest_factor n) Hcut.
399 apply transitive_divides m (smallest_factor n).
401 apply divides_smallest_factor_n.
402 apply trans_lt ? (S O). simplify. apply le_n. exact H.
403 apply lt_smallest_factor_to_not_divides.
404 exact H.assumption.assumption.assumption.
406 apply trans_lt O (S O).
408 apply lt_SO_smallest_factor.
413 theorem prime_to_smallest_factor: \forall n. prime n \to
414 smallest_factor n = n.
415 intro.apply nat_case n.intro.apply False_ind.apply not_prime_O H.
416 intro.apply nat_case m.intro.apply False_ind.apply not_prime_SO H.
419 (S O) < (S(S m1)) \land
420 (\forall m:nat. divides m (S(S m1)) \to (S O) < m \to m = (S(S m1))) \to
421 smallest_factor (S(S m1)) = (S(S m1)).
422 intro.elim H.apply H2.
423 apply divides_smallest_factor_n.
424 apply trans_lt ? (S O).simplify. apply le_n.assumption.
425 apply lt_SO_smallest_factor.
429 (* a number n > O is prime iff its smallest factor is n *)
430 definition primeb \def \lambda n:nat.
432 [ O \Rightarrow false
435 [ O \Rightarrow false
436 | (S q) \Rightarrow eqb (smallest_factor (S(S q))) (S(S q))]].
439 theorem example4 : primeb (S(S(S O))) = true.
440 normalize.reflexivity.
443 theorem example5 : primeb (S(S(S(S(S(S O)))))) = false.
444 normalize.reflexivity.
447 theorem example6 : primeb (S(S(S(S((S(S(S(S(S(S(S O)))))))))))) = true.
448 normalize.reflexivity.
451 theorem example7 : primeb (S(S(S(S(S(S((S(S(S(S((S(S(S(S(S(S(S O))))))))))))))))))) = true.
452 normalize.reflexivity.
455 theorem primeb_to_Prop: \forall n.
457 [ true \Rightarrow prime n
458 | false \Rightarrow \not (prime n)].
460 apply nat_case n.simplify.intro.elim H.apply not_le_Sn_O (S O) H1.
461 intro.apply nat_case m.simplify.intro.elim H.apply not_le_Sn_n (S O) H1.
464 match eqb (smallest_factor (S(S m1))) (S(S m1)) with
465 [ true \Rightarrow prime (S(S m1))
466 | false \Rightarrow \not (prime (S(S m1)))].
467 apply eqb_elim (smallest_factor (S(S m1))) (S(S m1)).
468 intro.change with prime (S(S m1)).
470 apply prime_smallest_factor_n.
471 simplify.apply le_S_S.apply le_S_S.apply le_O_n.
472 intro.change with \not (prime (S(S m1))).
473 change with prime (S(S m1)) \to False.
475 apply prime_to_smallest_factor.
479 theorem primeb_true_to_prime : \forall n:nat.
480 primeb n = true \to prime n.
483 [ true \Rightarrow prime n
484 | false \Rightarrow \not (prime n)].
486 apply primeb_to_Prop.
489 theorem primeb_false_to_not_prime : \forall n:nat.
490 primeb n = false \to \not (prime n).
493 [ true \Rightarrow prime n
494 | false \Rightarrow \not (prime n)].
496 apply primeb_to_Prop.
499 theorem decidable_prime : \forall n:nat.decidable (prime n).
500 intro.change with (prime n) \lor \not (prime n).
503 [ true \Rightarrow prime n
504 | false \Rightarrow \not (prime n)] \to (prime n) \lor \not (prime n).
505 apply Hcut.apply primeb_to_Prop.
506 elim (primeb n).left.apply H.right.apply H.
509 theorem prime_to_primeb_true: \forall n:nat.
510 prime n \to primeb n = true.
512 cut match (primeb n) with
513 [ true \Rightarrow prime n
514 | false \Rightarrow \not (prime n)] \to ((primeb n) = true).
515 apply Hcut.apply primeb_to_Prop.
516 elim primeb n.reflexivity.
517 absurd (prime n).assumption.assumption.
520 theorem not_prime_to_primeb_false: \forall n:nat.
521 \not(prime n) \to primeb n = false.
523 cut match (primeb n) with
524 [ true \Rightarrow prime n
525 | false \Rightarrow \not (prime n)] \to ((primeb n) = false).
526 apply Hcut.apply primeb_to_Prop.
528 absurd (prime n).assumption.assumption.